The real issue here is that federal and state governments didn't think through the logistics of distributing vaccines even though it has been many months since we've known a vaccine would be coming.
Where I live, there are about 10 different places that vaccinate, between county governments, pharmacies and hospitals. All of them have separate registration systems. Meanwhile, appointments go quickly and appointment availability is often announced on Twitter, which is probably isn't the best place for the 65+ crowd.
What they needed to do starting in December is create a state-wide (or at least county-wide) registration system. Essentially model it as a priority queue with different buckets:
1) covid ward healthcare workers
2) other front line healthcare workers
3) 65+
4) preexisting conditions
5) etc.
You can disagree about which buckets go first, but that's not my point here. This should have been setup and publicized well before the vaccine rollout started. Now, you get a sense of exactly who wants to be vaccinated, so you don't get wasted doses when 45% of a nursing home's staff doesn't want the shot.
You also ask people how far they would be willing to drive for a shot. I know people who drove a few hours. I know others who didn't want to drive more than 20 minutes. So it's relevant. You also ask times of day that people are willing to be vaccinated and if they want to be notified if earlier slots open up.
Once vaccines are actually available, you have a central queue and you pick randomly within each priority group. So let's say you are up to 65+ adults in the general population (which is a large group). 1000 appointments open up at a hospital in zip code 12345. You send out a text message and email to 1400 (since not all 1400 are going to respond to the text message) randomly selected people in zip code 12345 or who are willing to drive there. For some elderly people, you might need to make phone calls. You give people some time to respond and if they don't, then add them back to the pool and send messages to more people until you get all slots filled.
There are thousands of people in any major city who would drop what they are doing and run out to get the vaccine on a moment's notice. With a better system in place, this should never have fallen on the doctor in the first place.